Is there a context in which you found Fr(X) mentioned? Was Fr(X) claimed to have some particular properties? As you say, this is an unusual notation, but some guesses based on the fact that X is a Euclidean subspace are:

1. A frame of X viewed as a vector space. If X is embedded in another vector space V, then the a basis on V, restricted to X, would be naturally viewed as a collection of linearly dependent vectors that span X.

2. Fr(X) could be a space defined in a similar way to the Grassmannian; perhaps as the space of all copies of X in the original space.

3. With X viewed as a vector space that is a subspace of a manifold M, we might imagine using X to construct a vector bundle over M. If the notation is abused to also call this bundle X, then Fr(X) could be the frame bundle associated to this vector bundle.

I'm sure there are less obvious possibilities. It could even be that the context is Frechet spaces or manifolds.